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Lattice Attacks Against Elliptic-Curve Signatures with Blinded Scalar Multiplication

  • Dahmun Goudarzi
  • Matthieu Rivain
  • Damien Vergnaud
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10532)

Abstract

Elliptic curve cryptography is today the prevailing approach to get efficient public-key cryptosystems and digital signatures. Most of elliptic curve signature schemes use a nonce in the computation of each signature and the knowledge of this nonce is sufficient to fully recover the secret key of the scheme. Even a few bits of the nonce over several signatures allow a complete break of the scheme by lattice-based attacks. Several works have investigated how to efficiently apply such attacks when partial information on the nonce can be recovered through side-channel attacks. However, these attacks usually target unprotected implementation and/or make ideal assumptions on the recovered information, and it is not clear how they would perform in a scenario where common countermeasures are included and where only noisy information leaks via side channels. In this paper, we close this gap by applying such attack techniques against elliptic-curve signature implementations based on a blinded scalar multiplication. Specifically, we extend the famous Howgrave-Graham and Smart lattice attack when the nonces are blinded by the addition of a random multiple of the elliptic-curve group order or by a random Euclidean splitting. We then assume that noisy information on the blinded nonce can be obtained through a template attack targeting the underlying scalar multiplication and we show how to characterize the obtained likelihood scores under a realistic leakage assumption. To deal with this scenario, we introduce a filtering method which given a set of signatures and associated likelihood scores maximizes the success probability of the lattice attack. Our approach is backed up with attack simulation results for several signal-to-noise ratio of the exploited leakage.

Notes

Acknowledgments

The authors were supported in part by the French ANR JCJC ROMAnTIC project (ANR-12-JS02-0004).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Dahmun Goudarzi
    • 1
    • 2
  • Matthieu Rivain
    • 1
  • Damien Vergnaud
    • 2
  1. 1.CryptoExpertsParisFrance
  2. 2.ENS, CNRS, Inria and PSL Research UniversityParisFrance

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